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\nopagenumbers

\vglue -10pt



\cl {\bf About  Space Curves of Constant Torsion }
\lf
\cl { See also: About Space Curves of Constant Curvature  }
\lf
\cl{\sc  Definition via Differential Equations}

\Lf
Most Space Curves that 3DXM can exhibit are given in terms of explicit formulas
or explicit geometric constructions. In ``About Space Curves of Constant Curvature''
we explain how {\it curvature } and  {\it torsion } of a space curve are defined. The
definition immediately translates into a construction of the curve from curvature
and torsion via the following {\it differential equations}, the famous
$$\eqalign{
&\hbox{\it Frenet-Serret Equations: } \cr
&\dot e_1(t) := \hskip1cm\kappa(t)\cdot e_2(t), \cr
&\dot e_2(t) := -\kappa(t)\cdot e_1(t) - \tau(t)\cdot e_3(t), \cr
&\dot e_3(t):= \hskip1cm\tau(t)\cdot e_2(t).
}$$
For given continuous functions $\kappa,\tau$ these differential equations
have --- for given orthonormal initial values --- unique orthonormal solutions
$\{e_1(t), e_2(t), e_3(t) \}$. The curve $c(t) := \int ^t e_1(s)ds$ is then parametrized
by arc length and has the given curvature functions $\kappa, \tau$.
\LF
The simplest curves in the plane are straight lines and circles, curves of constant
curvature. It is therefore natural to discuss also space curves of constant curvature.
In 3DXM we illustrate these by finding closed examples in the following family:

$ \kappa(t) := aa,$

$\tau(t):= bb + cc\cdot\sin(t) + dd\cdot\sin(2t) + ee\cdot\sin(3t)$. 
\LF
To understand the Frenet-Serret equations better one can also study other
special cases. Experimentation shows that the following curves of constant torsion

$\kappa(t):= bb + cc\cdot\cos(f\!f\cdot t) + dd\cdot\cos(2f\!f\cdot  t) +$ 
\lf
\phantom{$\kappa(t):=$ aaa} $ee\cdot\cos(3f\!f\cdot t)$

$\tau(t) = aa$
\lf
have an amusingly strong change of shape as one changes the parameters. Again we
look for closed examples with the help of symmetries. Note that $180^\circ$ rotations
around the principal normals $e_2(t)$ at $ t/f\!f = k\pi, k\in \BZ$ are isometries of the
curves. At $t/f\!f = \pi/2 + k\pi, k\in \BZ$ the $180^\circ$ rotations around the other
normal vector of the frame, $e_3(t)$, are also isometries of the space curve. This
allows us to formulate the {\it closing condition}:
\lf
If the normals $e_2(0)$ at $c(0)$, $e_3(\pi\!\cdot\! f\!f/2)$ at $c(\pi\!\cdot\! f\!f/2)$ intersect 
and if their angle is a rational multiple of $\pi$ then the space curve closes up.
Numerically one can use the parameter $cc$ to keep the angle constant, e.g. at
$\pi/3, \pi/4$, and then use $aa$ to let the normals intersect. There are many closed 
solutions. Typically they look like a collection of bed springs which are joint by
fairly straight pieces. If one allows these bed springs to have many turns then the
closing values of $aa$ and $cc$ are almost equidistant. The default morph of 3DXM
shows this, it contains two closed and three approximately closed curves which
are made of {\it three} bed springs with an increasing number of turns. It is easy to 
extend this family to springs with more turns, but one can also find all the small 
values, down to just one half turn for each spring. --- We found no closed curves 
made of only {\it two springs}. 
\lf
Here is a list of numerically closed curves:
\LF
Curves with 3-fold symmetry, $f\!f=0.208$,
$$\matrix{aa, &0.178632213, &0.284031845, &0.417033334, 
    \cr         cc,  &0.2874008,   &0.90658882,  &2.19234962, 
    \cr         aa, &0.513441035,  &0.59263462, &0.628044,
    \cr        cc,  &3.489480574,  &4.7901189, &5.4411264,
    \cr        aa, & 0.661324546, &0.69281176, &0.7227614
    \cr        cc, & 6.09244336, & 6.7440016, & 7.39575343
    }$$
\noindent
Curves with 4-fold symmetry, $f\!f=0.23$,
$$\matrix{aa, &0.2137654757, &0.3704887, &0.479019355, 
   \cr           cc, &0.234123448, &0.89640923, &1.59595534,
   \cr           aa, &0.56642393, &0.6414483533, &0.7081321561,
   \cr           cc, &2.30473675, &3.01756515691, &3.732639742,
   \cr           aa, &0.76871766, &0.8246012, &0.87671763
   \cr           cc, &4.449136, &5.1666082, &5.8847911
   }$$
 \noindent
Curve with 5-fold symmetry, $f\!f=0.2324$, \lf
$    aa = 0.73855871446286,\ \   cc = 2.96466
$
\LF
If ones does not begin with the differential equation but starts from the curve, then
one cannot define the torsion at points where the curvature vanishes. This problem
is caused by the use of the Frenet frame. Another frame is suggested by a mechanical
consideration: If a massive sphere would move along the space curve (imagine the
space curve as a wire and the sphere with a hole through which the wire slides
without friction) then inertia would make the sphere avoid unnecessary rotations
around the wire. In other words: A frame which is attached to the sphere so that it
is normal to the wire remains normal and the derivatives of the normal vectors have
{\it no normal components}. Such frames are called ``parallel as normal vectors'',
or simply ``parallel frames''.
In 3DXM one can choose {\tt Parallel Frame} in the Action Menu . Now {\tt Show Curve
as Tube} illustrates the behaviour of the chosen frame. In particular the torus knots
show how the parallel frames avoid ``unnecessary'' rotations which the Frenet frames
must make.
\LF
An advantage of 
such parallel frames is that they neither require to assume more than {\it two} 
continuous derivatives of the curve nor that 
$\kappa$ never vanishes---even straight lines are not exceptional curves if one 
works with these frames. Let $\phi(t)$ be an antiderivative of the torsion function,
i.e., $\dot\phi(t) = \tau(t)$. Then the differential equation that determines this frame 
has the following simple form:
$$\eqalign{
&\hbox{\it Frenet-Serret Equations for Parallel Frames: } \cr
&\dot e_1(t) := \hskip5mm \kappa(t)\cos(\phi(t))\cdot e_2(t) +\kappa(t)\sin(\phi(t))\cdot e_3(t) \cr
&\dot e_2(t) :=  -\kappa(t)\cos(\phi(t))\cdot e_1(t) \cr
&\dot e_3(t):=  -\kappa(t)\sin(\phi(t))\cdot e_1(t).
}$$

\bye

ff = 0.208;aa=[aa, 0.178632213];cc=[cc, 0.2874008];aa=[aa, 0.284031845];cc=[cc, 0.90658882];aa=[aa, 0.417033334];cc=[cc, 2.19234961567];aa=[aa, 0.513441035];cc=[cc, 3.489480574];aa=[aa, 0.59263462];cc=[cc, 4.7901189];aa=[aa, 0.628044];cc=[cc, 5.4411264];aa=[aa, 0.661324546];cc=[cc, 6.09244336];aa=[aa, 0.69281176];cc=[cc, 6.7440016];aa=[aa, 0.7227614];cc=[cc, 7.39575343];
aa = 0.731272348;  cc = 6.736836375;  ff = 0.23; % 3-fold closed
aa = 0.76871766;  cc = 4.449136;  ff = 0.23; % 4-fold closed
aa = 0.73855871446286;  cc = 2.96466;  ff = 0.2324; % 5-fold closed
\lf\phantom{.}\hskip1cm